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<h2 class="hd hd-2 unit-title">Quantum cryptography I</h2>
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Quantum cryptography
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<p>
In this three-part problem, we look at a specific example of BB84 protocol, which is a protocol for quantum cryptography proposed by Bennet and Brassard in 1984. </p>
<p>
We assume that there is no disturbance on the quantum state being sent from Alice to Bob due to noise effects. </p>
<p>
Suppose Alice would like to send an [mathjaxinline]n[/mathjaxinline] bit string [mathjaxinline]a[/mathjaxinline], which should serve as a secret key. Alice now generates an [mathjaxinline]n[/mathjaxinline] bit string [mathjaxinline]b[/mathjaxinline] by randomly sampling each bit. She encodes each bit of [mathjaxinline]a[/mathjaxinline] into quantum state by the following way. Let [mathjaxinline]a_ i[/mathjaxinline] and [mathjaxinline]b_ i[/mathjaxinline] denote [mathjaxinline]i[/mathjaxinline]th bit of string [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline]. If [mathjaxinline]b_ i=0[/mathjaxinline], she encodes [mathjaxinline]a_ i[/mathjaxinline] into [mathjaxinline]|{a_ i}\rangle[/mathjaxinline]. If, on the other hand, [mathjaxinline]b_ i=1[/mathjaxinline], she encodes [mathjaxinline]a_ i[/mathjaxinline] into [mathjaxinline]H|{a_ i}\rangle[/mathjaxinline]. Let us call this [mathjaxinline]n[/mathjaxinline]-qubit quantum state [mathjaxinline]|{\psi }\rangle[/mathjaxinline]. </p>
<ul class="itemize">
<li>
<p>
Suppose [mathjaxinline]a=011011000010[/mathjaxinline] and [mathjaxinline]b=110100101100[/mathjaxinline]. What are the states on the 2nd, 5th and 7th qubit of [mathjaxinline]|{\psi }\rangle[/mathjaxinline]? Answer using the ket notation with [mathjaxinline]\{ |{0}\rangle ,|{1}\rangle \}[/mathjaxinline] basis. </p>
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2nd qubit:<br/><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_Quantum_cryptography_2_1" class="text-input-dynamath capa_inputtype textline">
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5th qubit:<br/><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_Quantum_cryptography_3_1" class="text-input-dynamath capa_inputtype textline">
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7th qubit:<br/><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_Quantum_cryptography_4_1" class="text-input-dynamath capa_inputtype textline">
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<h2 class="hd hd-2 unit-title">Quantum cryptography II</h2>
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Quantum cryptography
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<p>
In this three-part problem, we look at a specific example of BB84 protocol, which is a protocol for quantum cryptography proposed by Bennet and Brassard in 1984. </p>
<p>
We assume that there is no disturbance on the quantum state being sent from Alice to Bob due to noise effects. </p>
<p>
Suppose Alice would like to send an [mathjaxinline]n[/mathjaxinline] bit string [mathjaxinline]a[/mathjaxinline], which should serve as a secret key. Alice now generates an [mathjaxinline]n[/mathjaxinline] bit string [mathjaxinline]b[/mathjaxinline] by randomly sampling each bit. She encodes each bit of [mathjaxinline]a[/mathjaxinline] into quantum state by the following way. Let [mathjaxinline]a_ i[/mathjaxinline] and [mathjaxinline]b_ i[/mathjaxinline] denote [mathjaxinline]i[/mathjaxinline]th bit of string [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline]. If [mathjaxinline]b_ i=0[/mathjaxinline], she encodes [mathjaxinline]a_ i[/mathjaxinline] into [mathjaxinline]|{a_ i}\rangle[/mathjaxinline]. If, on the other hand, [mathjaxinline]b_ i=1[/mathjaxinline], she encodes [mathjaxinline]a_ i[/mathjaxinline] into [mathjaxinline]H|{a_ i}\rangle[/mathjaxinline]. Let us call this [mathjaxinline]n[/mathjaxinline]-qubit quantum state [mathjaxinline]|{\psi }\rangle[/mathjaxinline]. </p>
<ul class="itemize">
<li>
<p>
Now, Alice sends [mathjaxinline]|{\psi }\rangle[/mathjaxinline] to Bob. We first assume that there is no eavesdropper trying to steal the information. Bob also generates a random bit string [mathjaxinline]b'[/mathjaxinline] and measures [mathjaxinline]i[/mathjaxinline]th qubit of [mathjaxinline]|{\psi }\rangle[/mathjaxinline] with 1)[mathjaxinline]\{ |{0}\rangle ,|{1}\rangle \}[/mathjaxinline] basis if [mathjaxinline]b'_ i=0[/mathjaxinline] and 2)[mathjaxinline]\{ |{+}\rangle ,|{-}\rangle \}[/mathjaxinline] basis if [mathjaxinline]b'_ i=1[/mathjaxinline]. Let [mathjaxinline]a'[/mathjaxinline] be a string that stores the result of the measurement results obtained by Bob. </p>
<p>
Alice and Bob then publicly announce [mathjaxinline]b[/mathjaxinline] and [mathjaxinline]b'[/mathjaxinline]. They only keep the bits of [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]a'[/mathjaxinline] satisfying [mathjaxinline]b_ i=b'_ i[/mathjaxinline] and discard the other bits. Let [mathjaxinline]a_ s[/mathjaxinline] and [mathjaxinline]a'_ s[/mathjaxinline] be the remaining bit strings. Suppose [mathjaxinline]b'=101100010110[/mathjaxinline]. What string [mathjaxinline]a'_ s[/mathjaxinline] will Bob end up having after discarding some bits in [mathjaxinline]a'[/mathjaxinline]? Please make the string a vector to answer. For instance, you should input <tt class="ttfamily">[0,0,1]</tt> to input string 001. </p>
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<h2 class="hd hd-2 unit-title">Quantum cryptography III</h2>
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<h3 class="hd hd-3 problem-header" id="problem_Quantum_cryptographyx-problem-title" aria-describedby="block-v1:MITx+8.370.3x+1T2018+type@problem+block@problem_Quantum_cryptographyx-problem-progress" tabindex="-1">
Quantum cryptography
</h3>
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<p>
In this three-part problem, we look at a specific example of BB84 protocol, which is a protocol for quantum cryptography proposed by Bennet and Brassard in 1984. </p>
<p>
We assume that there is no disturbance on the quantum state being sent from Alice to Bob due to noise effects. </p>
<p>
Suppose Alice would like to send an [mathjaxinline]n[/mathjaxinline] bit string [mathjaxinline]a[/mathjaxinline], which should serve as a secret key. Alice now generates an [mathjaxinline]n[/mathjaxinline] bit string [mathjaxinline]b[/mathjaxinline] by randomly sampling each bit. She encodes each bit of [mathjaxinline]a[/mathjaxinline] into quantum state by the following way. Let [mathjaxinline]a_ i[/mathjaxinline] and [mathjaxinline]b_ i[/mathjaxinline] denote [mathjaxinline]i[/mathjaxinline]th bit of string [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline]. If [mathjaxinline]b_ i=0[/mathjaxinline], she encodes [mathjaxinline]a_ i[/mathjaxinline] into [mathjaxinline]|{a_ i}\rangle[/mathjaxinline]. If, on the other hand, [mathjaxinline]b_ i=1[/mathjaxinline], she encodes [mathjaxinline]a_ i[/mathjaxinline] into [mathjaxinline]H|{a_ i}\rangle[/mathjaxinline]. Let us call this [mathjaxinline]n[/mathjaxinline]-qubit quantum state [mathjaxinline]|{\psi }\rangle[/mathjaxinline]. </p>
<ul class="itemize">
<li>
<p>
Let the length of [mathjaxinline]a_ s[/mathjaxinline] and [mathjaxinline]a'_ s[/mathjaxinline] be [mathjaxinline]k[/mathjaxinline]. Alice generates a [mathjaxinline]k[/mathjaxinline] bit random string [mathjaxinline]c[/mathjaxinline] and publicly shares it with Bob. They pick the bits of [mathjaxinline]a_ s[/mathjaxinline] and [mathjaxinline]a'_ s[/mathjaxinline] for which [mathjaxinline]c[/mathjaxinline] has 1 and compare them to each other. Let [mathjaxinline]t[/mathjaxinline] be the number of the bits on which the two strings match. What is the expectation value of [mathjaxinline]t[/mathjaxinline]? Answer in terms of [mathjaxinline]k[/mathjaxinline]. </p>
<p>
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</p>
</li>
<li>
<p>
Now suppose Eve tries to steal the secret key. She intercepts [mathjaxinline]|{\psi }\rangle[/mathjaxinline], makes a measurement on each bit with randomly chosen basis between [mathjaxinline]\{ |{0}\rangle ,|{1}\rangle \}[/mathjaxinline] and [mathjaxinline]\{ |{+}\rangle ,|{-}\rangle \}[/mathjaxinline], and sends the postmeasurement state to Bob. What is the expectation value of [mathjaxinline]t[/mathjaxinline] in this case? Answer in terms of [mathjaxinline]k[/mathjaxinline]. </p>
<p>
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<div class="unanswered ">
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</p>
</li>
<li>
<p>
Suppose Eve instead makes a measurement on each bit with basis [mathjaxinline]\{ |{H}\rangle ,|{\bar{H}}\rangle \}[/mathjaxinline] where [mathjaxinline]|{H}\rangle[/mathjaxinline] and [mathjaxinline]|{\bar{H}}\rangle[/mathjaxinline] are the eigenstate of the Hadamard gate with eigenvalue +1 and -1 respectively, and sends the postmeasurement state to Bob. What is the expectation value of [mathjaxinline]t[/mathjaxinline] in this case? Answer in terms of [mathjaxinline]k[/mathjaxinline]. </p>
<p>
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